Combinations Calculator

Combinations are solved using the following formula: ${}_n{C}_r=\frac{n!}{r!(n-r)!}$. In this formula n is the number of objects in the set and r is the size of the subsets, and ! is the factorial symbol. For more on factorials, check out our factorial calculator. Let's try solving the following combination. $${}_{18}{C}_4$$Our n is 18 and our r is 4. In English, this would be the same as saying "18 choose 4" or selecting 4 items from a group of 18. Let's plug our numbers into the equation. $$\frac{18!}{4!(18-4)!}$$We can now start solving. At this point, you could just use a calculator to evaluate the equation. However, if one isn't handy, there are some things we can do to simplify the problem. To start, complete the subtraction in the denominator. $$\frac{18!}{4!14!}$$Remember that factorials multiply a number by every number between it and 0. With that in mind, we can expand the numerator. $$\frac{18\times 17\times 16\times 15\times 14!}{4!14!}$$Now, we can cancel the 14! out of the division, since it appears in both the numerator and denominator. $$\frac{18\times 17\times 16\times 15}{4!}$$Now we have greatly reduced the amount of multiplication we must do to solve the expression. Evaluating it yields 3060, meaning there are 3060 ways to choose 4 from 18.